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Data publicarii: 16 Octombrie, 2008

IDENTITATI TRIGONOMETRICE-teorie

1)\; {\sin ^2}{a}+{\cos^2}{a}=1,\forall{a}\in{\mathbb{R}}.1)\; {\sin ^2}{a}+{\cos^2}{a}=1,\forall{a}\in{\mathbb{R}}.

2)\;\sin(-x)=-\sin{x} ,\forall{x}\in{\mathbb{R}}.2)\;\sin(-x)=-\sin{x} ,\forall{x}\in{\mathbb{R}}.

3)\;\cos{(-x)}=\cos{x} ,\forall{x}\in{\mathbb{R}}.3)\;\cos{(-x)}=\cos{x} ,\forall{x}\in{\mathbb{R}}.

4)\;{tgx} = \frac{\sin{x}}{\cos{x}},\forall{x}\in{\mathbb{R}}\setminus{\begin{Bmatrix} (2k+1)\frac{\pi}{2}|k\in{\mathbb{Z}}\end{Bmatrix}}.4)\;{tgx} = \frac{\sin{x}}{\cos{x}},\forall{x}\in{\mathbb{R}}\setminus{\begin{Bmatrix} (2k+1)\frac{\pi}{2}|k\in{\mathbb{Z}}\end{Bmatrix}}.

5)\;{ctgx} = \frac{\cos{x}}{\sin{x}},\forall{x}\in{\mathbb{R}}\setminus\{{k\pi}|k\in{\mathbb{Z}}\}.5)\;{ctgx} = \frac{\cos{x}}{\sin{x}},\forall{x}\in{\mathbb{R}}\setminus\{{k\pi}|k\in{\mathbb{Z}}\}.

6)\;{tg(-x)} = {- tgx}, \forall{x}\in{\mathbb{R}}\setminus{\begin{Bmatrix}(2k+1)\frac{\pi}{2}|k\in{\mathbb{Z}}\end{Bmatrix}}.6)\;{tg(-x)} = {- tgx}, \forall{x}\in{\mathbb{R}}\setminus{\begin{Bmatrix}(2k+1)\frac{\pi}{2}|k\in{\mathbb{Z}}\end{Bmatrix}}.

7)\;{ctg(-x)}={- ctgx},\forall{x}\in{\mathbb{R}}\setminus{\begin{Bmatrix}k\pi|k\in{\mathbb{Z}}\end{Bmatrix}}.7)\;{ctg(-x)}={- ctgx},\forall{x}\in{\mathbb{R}}\setminus{\begin{Bmatrix}k\pi|k\in{\mathbb{Z}}\end{Bmatrix}}.

8)\;{secx}=\frac{1}{cosx},\;\forall{x}\in{\mathbb{R}}\setminus{\begin{Bmatrix}(2k+1)\frac{\pi}{2}|k\in{\mathbb{Z}}\end{Bmatrix}}.8)\;{secx}=\frac{1}{cosx},\;\forall{x}\in{\mathbb{R}}\setminus{\begin{Bmatrix}(2k+1)\frac{\pi}{2}|k\in{\mathbb{Z}}\end{Bmatrix}}.

9)\;{cosecx}=\frac{1}{sinx},\;\forall{x}\in{\mathbb{R}}\setminus{\begin{Bmatrix}k\pi|k\in{\mathbb{Z}}\end{Bmatrix}}.9)\;{cosecx}=\frac{1}{sinx},\;\forall{x}\in{\mathbb{R}}\setminus{\begin{Bmatrix}k\pi|k\in{\mathbb{Z}}\end{Bmatrix}}.  

10)\;\cos{(a+b)}=\cos{a}\cos{b}-\sin{a}\sin{b}, \forall{a,b}\in{\mathbb{R}}.10)\;\cos{(a+b)}=\cos{a}\cos{b}-\sin{a}\sin{b}, \forall{a,b}\in{\mathbb{R}}.

11)\;\cos{2a}={{\cos}^2}{a}-{{\sin}^2}{a} = 2{{\cos}^2}{a} - 1 = 1 - 2{{\sin}^2}{a},\forall{a}\in{\mathbb{R}}.11)\;\cos{2a}={{\cos}^2}{a}-{{\sin}^2}{a} = 2{{\cos}^2}{a} - 1 = 1 - 2{{\sin}^2}{a},\forall{a}\in{\mathbb{R}}.

12)\;\cos{3a} =\ cos{a}(4{\cos^2}{a}-3),\forall{a}\in{\mathbb{R}}.12)\;\cos{3a} =\ cos{a}(4{\cos^2}{a}-3),\forall{a}\in{\mathbb{R}}.

13)\;\cos{(a-b)}=\cos{a}\cos{b}+\sin{a}\sin{b}, \forall{a,b}\in{\mathbb{R}}.13)\;\cos{(a-b)}=\cos{a}\cos{b}+\sin{a}\sin{b}, \forall{a,b}\in{\mathbb{R}}.

14)\;\cos{(\frac{\pi}{2}-x)}=\sin{x},\forall{x}\in{\mathbb{R}}.14)\;\cos{(\frac{\pi}{2}-x)}=\sin{x},\forall{x}\in{\mathbb{R}}.

15)\;\cos{(\pi-x)}= -\ cos{x}, \forall{x}\in{\mathbb{R}}.15)\;\cos{(\pi-x)}= -\ cos{x}, \forall{x}\in{\mathbb{R}}.

16)\;\cos{(x+2k\pi)} =\ cos{x},\forall{x}\in{\mathbb{R}},\forall{k}\in{\mathbb{Z}}.16)\;\cos{(x+2k\pi)} =\ cos{x},\forall{x}\in{\mathbb{R}},\forall{k}\in{\mathbb{Z}}.

17)\;\sin{(a+b)} =\ sin{a}\cos{b} +\ cos{a}\sin{b},\forall{a,b}\in{\mathbb{R}}.17)\;\sin{(a+b)} =\ sin{a}\cos{b} +\ cos{a}\sin{b},\forall{a,b}\in{\mathbb{R}}.

18)\;\sin{2a} = 2\sin{a}\cos{a},\forall{a}\in{\mathbb{R}}.18)\;\sin{2a} = 2\sin{a}\cos{a},\forall{a}\in{\mathbb{R}}.

19)\;\sin{3a} =\ sin{a}(3 - 4{\sin^2}{a}), \forall{a}\in{\mathbb{R}}.19)\;\sin{3a} =\ sin{a}(3 - 4{\sin^2}{a}), \forall{a}\in{\mathbb{R}}.

20)\;\sin{(a-b)}= \sin{a}\cos{b} - \cos{a}\sin{b}, \forall{a,b}\in{\mathbb{R}}.20)\;\sin{(a-b)}= \sin{a}\cos{b} - \cos{a}\sin{b}, \forall{a,b}\in{\mathbb{R}}.

21)\;\sin{(\frac{\pi}{2}-x)} = \cos{x},\forall{x}\in{\mathbb{R}}.21)\;\sin{(\frac{\pi}{2}-x)} = \cos{x},\forall{x}\in{\mathbb{R}}.

22)\;\sin{(\pi-x)} =\ sin{x},\forall{x}\in{\mathbb{R}}.22)\;\sin{(\pi-x)} =\ sin{x},\forall{x}\in{\mathbb{R}}.

23)\;\sin{(x + 2k\pi)} =\ sin{x}, \forall{x}\in{\mathbb{R}},\forall{k}\in{\mathbb{Z}}.23)\;\sin{(x + 2k\pi)} =\ sin{x}, \forall{x}\in{\mathbb{R}},\forall{k}\in{\mathbb{Z}}.

24)\;{tg(x + k\pi)} = {tgx},\forall{x}\in{\mathbb{R}}\setminus{\begin{Bmatrix}(2k+1){\frac{\pi}{2}},\forall{k}\in{\mathbb{Z}}\end{Bmatrix}}.24)\;{tg(x + k\pi)} = {tgx},\forall{x}\in{\mathbb{R}}\setminus{\begin{Bmatrix}(2k+1){\frac{\pi}{2}},\forall{k}\in{\mathbb{Z}}\end{Bmatrix}}.

25)\;{ctg(x + k\pi)} = {ctgx},\forall{x}\in{\mathbb{R}}\setminus{\begin{Bmatrix}k\pi,\forall{k}\in{\mathbb{Z}}\end{Bmatrix}}.25)\;{ctg(x + k\pi)} = {ctgx},\forall{x}\in{\mathbb{R}}\setminus{\begin{Bmatrix}k\pi,\forall{k}\in{\mathbb{Z}}\end{Bmatrix}}.

26)\;{tg({a}\pm{b})} = \frac{{tga}\pm{tgb}}{1\mp{tga}{tgb}},\forall{a,b,{a}\pm{b}\in{\mathbb{R}}}\setminus{\begin{Bmatrix}(2k+1)\frac{\pi}{2},{k}\in{\mathbb{Z}}\end{Bmatrix}}.26)\;{tg({a}\pm{b})} = \frac{{tga}\pm{tgb}}{1\mp{tga}{tgb}},\forall{a,b,{a}\pm{b}\in{\mathbb{R}}}\setminus{\begin{Bmatrix}(2k+1)\frac{\pi}{2},{k}\in{\mathbb{Z}}\end{Bmatrix}}.

27)\;{tg2a} =\frac{2tga}{1-{{tg}^{2}}{a}}, \forall{a}\in{\mathbb{R}}\setminus{\begin{Bmatrix}(2k+1)\frac{\pi}{2}, k\in{\mathbb{Z}}\end{Bmatrix}}\cup\begin{Bmatrix}(2k+1)\frac{\pi}{4}, k\in{\mathbb{Z}}\end{Bmatrix}.27)\;{tg2a} =\frac{2tga}{1-{{tg}^{2}}{a}}, \forall{a}\in{\mathbb{R}}\setminus{\begin{Bmatrix}(2k+1)\frac{\pi}{2}, k\in{\mathbb{Z}}\end{Bmatrix}}\cup\begin{Bmatrix}(2k+1)\frac{\pi}{4}, k\in{\mathbb{Z}}\end{Bmatrix}.

28)\;\sin{a}=\frac{2tg{\frac{a}{2}}}{1+{tg}^{2}{\frac{a}{2}}}, \forall{a}\in{\mathbb{R}} \setminus{\begin{Bmatrix}(2k+1)\pi,k\in{\mathbb{Z}}\end{Bmatrix}}.28)\;\sin{a}=\frac{2tg{\frac{a}{2}}}{1+{tg}^{2}{\frac{a}{2}}}, \forall{a}\in{\mathbb{R}} \setminus{\begin{Bmatrix}(2k+1)\pi,k\in{\mathbb{Z}}\end{Bmatrix}}.

29)\;\cos{a} =\frac{1-{tg}^{2}{\frac{a}{2}}}{1+{tg}^{2}{\frac{a}{2}}}, \forall{a}\in{\mathbb{R}}\setminus{\begin{Bmatrix}(2k+1)\pi,k\in{\mathbb{Z}}\end{Bmatrix}}.29)\;\cos{a} =\frac{1-{tg}^{2}{\frac{a}{2}}}{1+{tg}^{2}{\frac{a}{2}}}, \forall{a}\in{\mathbb{R}}\setminus{\begin{Bmatrix}(2k+1)\pi,k\in{\mathbb{Z}}\end{Bmatrix}}.

30)\;{tga} =\frac{2tg\frac{a}{2}}{1-{{tg}^{2}}{\frac{a}{2}}}, \forall{a}\in{\mathbb{R}}\setminus{\begin{Bmatrix}(2k+1)\frac{\pi}{2}, k\in{\mathbb{Z}}\end{Bmatrix}}\cup\begin{Bmatrix}(2k+1)\pi, k\in{\mathbb{Z}}\end{Bmatrix}.30)\;{tga} =\frac{2tg\frac{a}{2}}{1-{{tg}^{2}}{\frac{a}{2}}}, \forall{a}\in{\mathbb{R}}\setminus{\begin{Bmatrix}(2k+1)\frac{\pi}{2}, k\in{\mathbb{Z}}\end{Bmatrix}}\cup\begin{Bmatrix}(2k+1)\pi, k\in{\mathbb{Z}}\end{Bmatrix}.

31)\;\sin{a} +\ sin{b} = 2\sin{\frac{a+b}{2}}\cos{\frac{a-b}{2}}, \forall{a,b}\in{\mathbb{R}}.31)\;\sin{a} +\ sin{b} = 2\sin{\frac{a+b}{2}}\cos{\frac{a-b}{2}}, \forall{a,b}\in{\mathbb{R}}.

32)\;\sin{a}-\ sin{b} = 2\sin{\frac{a-b}{2}}\cos{\frac{a+b}{2}}, \forall{a,b}\in{\mathbb{R}}.32)\;\sin{a}-\ sin{b} = 2\sin{\frac{a-b}{2}}\cos{\frac{a+b}{2}}, \forall{a,b}\in{\mathbb{R}}.

33)\;\cos{a} +\ cos{b} = 2\cos{\frac{a+b}{2}}\cos{\frac{a-b}{2}}, \forall{a,b}\in{\mathbb{R}}.33)\;\cos{a} +\ cos{b} = 2\cos{\frac{a+b}{2}}\cos{\frac{a-b}{2}}, \forall{a,b}\in{\mathbb{R}}.

34)\;\cos{a} -\ cos{b} = -2\sin{\frac{a+b}{2}}\sin{\frac{a-b}{2}}, \forall{a,b}\in{\mathbb{R}}.34)\;\cos{a} -\ cos{b} = -2\sin{\frac{a+b}{2}}\sin{\frac{a-b}{2}}, \forall{a,b}\in{\mathbb{R}}.

35)\;{tga}\pm{tgb}=\frac{\sin({a}\pm{b})}{{cosa}{cosb}}, \forall{a,b}\in{\mathbb{R}}\setminus{\begin{Bmatrix}(2k+1)\frac{\pi}{2}, k\in{\mathbb{Z}}\end{Bmatrix}}.35)\;{tga}\pm{tgb}=\frac{\sin({a}\pm{b})}{{cosa}{cosb}}, \forall{a,b}\in{\mathbb{R}}\setminus{\begin{Bmatrix}(2k+1)\frac{\pi}{2}, k\in{\mathbb{Z}}\end{Bmatrix}}.

36)\;{1 + cos{2a}} = 2{{\cos}^2}{a}\Leftrightarrow{|\cos{a}|} =\sqrt{\frac{1+\cos{2a}}{2}},\forall{a}\in{\mathbb{R}}.36)\;{1 + cos{2a}} = 2{{\cos}^2}{a}\Leftrightarrow{|\cos{a}|} =\sqrt{\frac{1+\cos{2a}}{2}},\forall{a}\in{\mathbb{R}}.

37)\;{1 - cos{2a}} = 2{{\sin}^2}{a}\Leftrightarrow{|\sin{a}|} =\sqrt{\frac{1-\cos{2a}}{2}},\forall{a}\in{\mathbb{R}}.37)\;{1 - cos{2a}} = 2{{\sin}^2}{a}\Leftrightarrow{|\sin{a}|} =\sqrt{\frac{1-\cos{2a}}{2}},\forall{a}\in{\mathbb{R}}.

38)\;tg{\frac{x}{2}}=\frac{sinx}{1+cosx}=\frac{1-cosx}{sinx},\;\forall{x}\in{\mathbb{R}}\setminus\{(2k+1){\pi}\},\;{k}\in{\mathbb{Z}}\}.38)\;tg{\frac{x}{2}}=\frac{sinx}{1+cosx}=\frac{1-cosx}{sinx},\;\forall{x}\in{\mathbb{R}}\setminus\{(2k+1){\pi}\},\;{k}\in{\mathbb{Z}}\}.

39)\;ctg{\frac{x}{2}}=\frac{1+cosx}{sinx}=\frac{sinx}{1-cosx},\;\forall{x}\in{\mathbb{R}}\setminus\{2k{\pi}\},\;{k}\in{\mathbb{Z}}\}.39)\;ctg{\frac{x}{2}}=\frac{1+cosx}{sinx}=\frac{sinx}{1-cosx},\;\forall{x}\in{\mathbb{R}}\setminus\{2k{\pi}\},\;{k}\in{\mathbb{Z}}\}.

40)\;{\sin{a}\cos{b}}=\frac{1}{2}\cdot[\sin{(a-b)}+\sin{(a+b)}],\forall{a,b}\in{\mathbb{R}}.40)\;{\sin{a}\cos{b}}=\frac{1}{2}\cdot[\sin{(a-b)}+\sin{(a+b)}],\forall{a,b}\in{\mathbb{R}}.

41)\;{\cos{a}\cos{b}}=\frac{1}{2}\cdot[\cos{(a-b)}+\cos{(a+b)}],\forall{a,b}\in{\mathbb{R}}.41)\;{\cos{a}\cos{b}}=\frac{1}{2}\cdot[\cos{(a-b)}+\cos{(a+b)}],\forall{a,b}\in{\mathbb{R}}.

42)\;{\sin{a}\sin{b}}=\frac{1}{2}\cdot[\cos{(a-b)}-\cos{(a+b)}],\forall{a,b}\in{\mathbb{R}}.42)\;{\sin{a}\sin{b}}=\frac{1}{2}\cdot[\cos{(a-b)}-\cos{(a+b)}],\forall{a,b}\in{\mathbb{R}}.

43)\;{\arcsin{x} +\ arccos{x}} =\frac{\pi}{2},\forall{x}\in{[-1,1]}.43)\;{\arcsin{x} +\ arccos{x}} =\frac{\pi}{2},\forall{x}\in{[-1,1]}.

44)\;{arctgx + arcctgx} =\frac{\pi}{2},\forall{x}\in{\mathbb{R}}.44)\;{arctgx + arcctgx} =\frac{\pi}{2},\forall{x}\in{\mathbb{R}}.

45)\;{\arcsin{(-x)}} = - {\arcsin}{x},\forall{x}\in{[-1,1]}.45)\;{\arcsin{(-x)}} = - {\arcsin}{x},\forall{x}\in{[-1,1]}.

46)\;{\arccos{(-x)}} = \pi -{\arccos}{x},\forall{x}\in{[-1,1]}.46)\;{\arccos{(-x)}} = \pi -{\arccos}{x},\forall{x}\in{[-1,1]}.

47)\;{arctg(-x)}=-{arctgx}, \forall{x}\in{\mathbb{R}}.47)\;{arctg(-x)}=-{arctgx}, \forall{x}\in{\mathbb{R}}.

48)\;{arcctg(-x)}=\pi -{arcctgx},\forall{x}\in{\mathbb{R}}.48)\;{arcctg(-x)}=\pi -{arcctgx},\forall{x}\in{\mathbb{R}}.

49)\;{{\sin}^2}{x} = \frac{{tg}^{2}{x}}{1+{tg}^{2}{x}},\forall{x}\in{\mathbb{R}}\setminus{\begin{Bmatrix}(2k+1)\frac{\pi}{2},k\in{\mathbb{Z}}\end{Bmatrix}}.49)\;{{\sin}^2}{x} = \frac{{tg}^{2}{x}}{1+{tg}^{2}{x}},\forall{x}\in{\mathbb{R}}\setminus{\begin{Bmatrix}(2k+1)\frac{\pi}{2},k\in{\mathbb{Z}}\end{Bmatrix}}.

50)\;{{\cos}^2}{x} = \frac{1}{1+{tg}^{2}{x}},\forall{x}\in{\mathbb{R}}\setminus{\begin{Bmatrix}(2k+1)\frac{\pi}{2},k\in{\mathbb{Z}}\end{Bmatrix}}.50)\;{{\cos}^2}{x} = \frac{1}{1+{tg}^{2}{x}},\forall{x}\in{\mathbb{R}}\setminus{\begin{Bmatrix}(2k+1)\frac{\pi}{2},k\in{\mathbb{Z}}\end{Bmatrix}}.

Valori remarcabile ale functiilor trigonometrice:

  

 0° 30° 45°60° 90°  180° 270° 360°
sin  0 \frac{1}{2}\frac{1}{2}  \frac{\sqrt{2}}{2}\frac{\sqrt{2}}{2}   \frac{\sqrt{3}}{2}\frac{\sqrt{3}}{2}  1 0 -1 0
cos 1 \frac{\sqrt{3}}{2}\frac{\sqrt{3}}{2}  \frac{\sqrt{2}}{2}\frac{\sqrt{2}}{2}    \frac{1}{2}\frac{1}{2}  0 -1 0 1
tg 0 \frac{\sqrt{3}}{3}\frac{\sqrt{3}}{3}  1 \sqrt{3}\sqrt{3}  / 0 / 0
ctg / \sqrt{3}\sqrt{3}  1 \frac{\sqrt{3}}{3}\frac{\sqrt{3}}{3}  0 / 0 /

 

 

 

 

 

         

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